Similarly to our previous case [1]},
we decompose \(\widetilde{R}_\varphi \)  with respect only to
the right or left component:
\(\widetilde{R}_\varphi &=& \bigoplus _n\int \prod _k \Gamma \left(\varphi \left(\frac{p_k}{P_1}\right)\right) \otimes dE_1(p_1)\otimes \cdots \otimes dE_1(p_n) \\&=& \bigoplus _m\int dE_1(p_1)\otimes \cdots \otimes dE_1(p_m )\otimes \prod _j \Gamma \left(\varphi \left(\frac{P_1}{p_j}\right)\right)\) 
